Method For Solving Deterministically Non-Linear Optimization Problems On Technical Constraints

ABSTRACT

The invention relates to a computer implemented method to optimize operation of a technical system by solving deterministically a nonlinear optimization problem implying technical constraints relating to technical parameters under which the said technical system operates, the technical parameters being of a number greater than 50, characterized in that the method comprises fractal geometric partitioning of a search space into a plurality of hyperspheres as a geometrical unitary pattern and wherein the hyperspheres partitioning said search space are overlapping; calculating a quality for each hypersphere; selecting from the plurality of hyperspheres, the hypersphere with the best quality; and further comprises determining an optimum solution of the said selected hypersphere, the solution comprising values of the technical parameters to be implemented in the said technical system.

FIELD OF THE DISCLOSURE

The present invention relates to the field of methods for solving optimization problems.

More specifically, the present invention relates to a computer implemented method to optimize operation of a technical system by solving deterministically a nonlinear optimization problem implying technical constraints relating to technical parameters under which the said technical system operates.

In particular, it finds advantageous application to optimize problems with a large number of technical parameters.

DESCRIPTION OF THE RELATED ART

This last decade, the complexity of optimization problems increased with the increase of the CPUs' power and the decrease of memory costs. Indeed, the appearance of clouds and other supercomputers provides the possibility to solve large scale problems. However, most of the deterministic and stochastic optimization methods see their performances go down with the increase of the number of parameters (dimension) of the problems. Evolutionary algorithms and other bio-inspired algorithms were widely used to solve large scale problems without much success.

The most efficient algorithms in literature are of stochastic nature, however this is a limiting factor when it comes to safety-critical applications where repeatability is important, as in image processing for example. Typically, in these cases, stochastic approach can be used only to improve the parameter settings of deterministic algorithms.

This stochastic nature of these algorithms makes their application in the industry complicated, because at each launch of the algorithm the provided result is not the same, therefore users generally content with a local solution (local optimum).

Moreover, the justification of an obtained solution can also be difficult, because the method used to deduce it is based on a complex stochastic search rather than on a deterministic approach. Another aspect is that the proof that a solution is optimal is not provided by stochastic heuristic approaches, not matters how much a solution might be close to an optimal solution.

The complexity of large scale problems or high dimensional non-convex functions comes from the fact that local minima (and maxima) are rare compared to another kind of point with zero gradient: a saddle point. Many classes of functions exhibit the following behavior: in low-dimensional spaces, local minima are common. In higher dimensional spaces, local minima are rare and saddle points are more common. For a function ƒ: R^(n)→R of this type, the expected ratio of the number of saddle points to local minima grows exponentially with n. For all these reasons, the use of gradient based algorithms is not recommended for large scale problems.

The search for the global optimum of an objective function is very complex, especially, when all variables have interaction with the decision. This interaction increases in the case of large scale problems, then, a large number of evaluations of the objective function is needed to find a good enough solution. On other terms, in the case of large-scale problems, the performance of optimization methods decreases drastically.

Such known methods which cannot solve accurately, or in a reasonable time, large-scale problems are for example the publication by IBM (U.S. Pat. No. 8,712,738) or the Random walk algorithm by Siemens—US 20080247646).

Moreover, said methods cover only the case of medical applications.

We also know methods using geometric fractal decomposition in order to model the search space and to reduce it. In those methods, at each iteration, the search space is divided into subspaces in order to trap the global optimum in a small interval.

It is known that in a reduced search space the use of metaheuristics allows reaching global solutions.

Thus, in 1999, Demirhan introduced a metaheuristic for global optimization based on geometric partitioning of the search space called FRACTOP (D. Ashlock and J. Schonfeld, “A fractal representation for real optimization” in 2007 IEEE Congress on Evolutionary Computation, September 2007, p. 87-94). The geometrical form used in the proposed method is the hypercube. A number of solutions are collected randomly from each subregion or using a metaheuristic such as Simulated Annealing (M. Dorigo, V. Maniezzo, and A. Colorni, “Positive feedback as a search strategy” Technical report 91-016, June 1991) or Genetic Algorithm (D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1st ed. Addison-Wesley Longman Publishing Co., Inc., 1989.). After that, a guidance system is set through fuzzy measures to lead the search to the promising subregions simultaneously and discard the other regions from partitioning.

The main characteristic of the decomposition procedure used in FRACTOP is that there are no overlaps, thus it does not visit the same local area more than once. This approach can be efficient for low dimension problems. However, the decomposition procedure generates a number of subregions, as a function of the dimension n. Hence, when n is higher the complexity of the partitioning method increases exponentially.

Another method using a representation based on the fractal geometry is proposed by Ashlock in 2007 called Multiple Optima Sierpinski Searcher (S. Kirkpatrick, “Optimization by simulated annealing: Quantitative studies” pp. 975-986, 1984). The fractal geometrical form chosen is the Sierpinski triangle generated using the chaos game which consists in moving a point repeatedly from a vertex to another selected randomly.

As for FRACTOP, this method requires 2^(n) generators for n dimensions to exploit the covering radius which makes this representation also suffers from the curse of dimensionality. Moreover, the search in Ashlock does not cover the entire feasible region.

Furthermore, obtained results by the approach of Ashlock demonstrate limitations of these approaches using fractals.

Therefore, these known methods of fractal decomposition, called interval optimization methods, cannot be applied to problems comprising a large number of parameters, as they are no longer effective.

Thus, there is a need to develop a method for solving optimization problems of large dimensions, and in particular non-linear problems, giving a more accurate solution by covering the whole search space and having an algorithm complexity manageable.

PRESENTATION OF THE INVENTION

An object of the invention is to propose a method for solving deterministically nonlinear optimization problems on technical constraints.

Yet another object of the invention is to determine a global optimum among the whole search space of candidate solutions. Another object of the invention is to propose a solution which allows reasonable computation times while still allowing reliability and robustness of results.

Another object of the invention is to propose a method for solving nonlinear optimization problems comprising a large number of technical parameters.

Thus, the invention proposes a computer implemented method to optimize operation of a technical system by solving deterministically a nonlinear optimization problem implying technical constraints relating to technical parameters under which the said technical system operates, said technical parameters being of a number greater than 50, characterized in that said method comprises:

-   -   a fractal geometric partitioning of a search space into a         plurality of hyperspheres as a geometrical unitary pattern and         wherein the hyperspheres partitioning said search space are         overlapping;     -   calculating a quality for each hypersphere;     -   selecting from the plurality of hyperspheres, the hypersphere         with the best quality; and further comprises determining an         optimum solution of the said selected hypersphere, the solution         comprising values of the technical parameters to be implemented         in the said technical system.

Advantageously, but optionally, the method according to the invention may also comprise at least one of the following characteristics:

-   -   the said method comprises the following steps:     -   a) initialization of a search hypersphere with a dimension D         equal to the number of parameters;     -   b) decomposing said search hypersphere into a plurality of         sub-hyperspheres, the said sub-hyperspheres overlapping each         other;     -   c) for each sub-hypersphere, calculation of a sub-hypersphere         quality and classification of the sub-hyperspheres as a function         of said calculated quality;     -   d) determination among the sub-hyperspheres of the         sub-hypersphere having the best quality;     -   e) until a first stopping criterion is reached, repetition of         steps b) to d), implemented on the basis of a search         hypersphere (H) corresponding to the sub-hypersphere having the         best quality as determined in the previous step;     -   f) when the first stopping criterion is reached, for each         sub-hypersphere resulting from the last implementation of step         b), determining and storing a solution of each sub-hypersphere         into a memory area of the computing unit;     -   g) until a second stopping criterion is reached, steps b) to e)         are implemented on the basis of a search hypersphere         corresponding to the following sub-hypersphere of the         classification determined in step c) following the penultimate         implementation of b); and     -   h) determining an optimum solution among the solutions stored in         the memory area and storing the values of the technical         parameters corresponding to this optimum solution;     -   the first stopping criterion is a maximum level of recursive         decomposition of the initial search hypersphere;     -   the second stopping criterion is a tolerance value related to         the problem to solve;     -   the second stopping criterion is reached when a solution is         stored for all the sub-hyperspheres from all the decomposition         levels of the fractal partitioning;     -   the step b) comprises the decomposition of the search         hypersphere in 2×D sub-hyperspheres;     -   the step a) of initialization comprises the determination of a         maximum level of recursive decomposition of the initial search         hypersphere;     -   the maximum level of recursive decomposition of the search         hypersphere is equal to 5;     -   the step c) further comprises storing ranked sub-hypersphere         into a memory area;     -   the step a) of initialization of the search hypersphere         comprises: determining the position of the center C and the         initial radius R of the search hypersphere, such as C=L+(U−L)/2,         R=(U−L)/2, wherein U is the upper bound and L is the lower bound         of a search space;     -   the sub-hyperspheres overlapping each other of step b) are         obtained by decomposing the search hypersphere into a plurality         of sub-hyperspheres, then applying a radius inflation factor of         the said sub-hyperspheres;     -   inflation factor corresponds to an increase of the radius of         sub-hyperspheres by a factor of at least 1.75;     -   the inflation factor corresponds to an increase of the radius of         sub-hyperspheres by a factor of at least 2.80;     -   the step d) of determination comprises:     -   for each sub-hypersphere,     -   (i) starting from the center {right arrow over (C)} of a         sub-hypersphere sH, generating two solutions {right arrow over         (S)}₁ and {right arrow over (S)}₂ along each dimension d of the         search space with

${\overset{\rightarrow}{S\; 1} = {\overset{\rightarrow}{C} + {\frac{r}{\sqrt{D}} \times \overset{\rightarrow}{e_{d}}}}},{\overset{\rightarrow}{S\; 2} = {\overset{\rightarrow}{C} - {\frac{r}{\sqrt{D}} \times \overset{\rightarrow}{e_{d}}}}},$

-   -   (ii) determining the quality q of the sub-hypersphere with q=max         {g1; g2; gc}     -   wherein:

${{g\; 1} = \frac{f\left( {\overset{\rightarrow}{S}}_{1} \right)}{{{\overset{\rightarrow}{S}}_{1} - {BSF}}}},{{g\; 2} = \frac{f\left( {\overset{\rightarrow}{S}}_{2} \right)}{{{\overset{\rightarrow}{S}}_{2} - {BSF}}}},{{{and}\mspace{14mu} {gc}} = \frac{f\left( \overset{\rightarrow}{C} \right)}{{{\overset{\rightarrow}{C}}^{k} - {BSF}}}},$

BSF corresponds to the position of the sub-hypersphere with the best quality determined so far, and ƒ({right arrow over (S)}₁), ƒ({right arrow over (S)}₂), ƒ({right arrow over (C)}) corresponds to the fitness of respectively {right arrow over (S)}₁, {right arrow over (S)}₂ and {right arrow over (C)};

-   -   a plurality of sub-hyperspheres is stored into a memory area;     -   the plurality of sub-hyperspheres is stored into a memory area         by the storing the position of the center C of the said         sub-hyperspheres;     -   the position of a plurality of sub-hyperspheres is computed from         a position of a sub-hypersphere stored into a memory area; and     -   the step d) of determining the sub-hypersphere having the best         quality and/or the step f) of determining and storing a solution         of each sub-hypersphere into a memory area is performed using         existing optimization software.

Furthermore, the invention relates to a computer program product having computer-executable instructions to enable a computer system to perform the method of any one of the characteristics described previously.

BRIEF DESCRIPTION OF DRAWINGS

Other characteristics, objects and advantages of the present invention will appear on reading the following detailed description and with reference to the accompanying drawings, given by way of non-limiting example and on which:

FIG. 1 shows some steps of a method for solving nonlinear optimization on technical constraints relating to technical parameters according to the invention;

FIGS. 2a and 2b show a representation of a search space by a set of hyperspheres according to the invention;

FIGS. 3a and 3b show a decomposition of an hypersphere by sub-hyperspheres and their inflation according to the invention;

FIG. 4 shows a performance comparison of the method according to the invention with known algorithms;

FIG. 5 shows a modeling of a hydroelectric power station;

FIG. 6a illustrates the determination of technical parameters of an image registration according to the invention;

FIG. 6b shows a convergence curve of errors obtained by applying the method according to the invention; and

FIGS. 7a to 7e show a distribution comparison of average ranks between the method according to the invention and known algorithms for set of dimensions respectively equal to 50, 100, 200, 500 and 1000.

DETAILED DESCRIPTION OF AT LEAST ONE EMBODIMENT OF THE INVENTION

FIG. 1 illustrates some steps of the method 100 for solving nonlinear optimization on technical constraints relating to technical parameters.

The said method 100 for solving optimization problems relies on a recursive decomposition of a search space E (the space of candidate solutions of an optimization problem) modeled by a hypersphere H into a given number of sub-hyperspheres sH that are, afterwards, enlarged in order to cover all the search space E.

Such recursive division of the search space E with fixed number of sub-hyperspheres represented, is called a fractal decomposition and the number of sub-hyperspheres sH inside a hypersphere H is called the fractal dimension.

Hence, in the following subsections, we will detail this strategy.

Decomposition of the Search Space by Hyperspheres

In a step 110, the search space E is approximated, at first, by an initial hypersphere H^(O). The choice of this geometry is motivated by the low complexity and its flexibility to cover all the search spaces. The initial hypersphere H⁰ is described using two parameters: position of its center C and its initial radius R that can be obtained using the following expressions:

C=L+(U−L)/2;

R=(U−L)/2;

where U is the upper bound and L is the lower bound of the whole search space E.

The hypersphere H⁰ is a sphere of D dimensions, D corresponding to the number of technical parameters of an optimization problem on technical constraints.

Then, in a step 120 this first hypersphere H⁰ is divided into a plurality of inferior sub-hyperspheres sH¹. The said plurality can be a multiple of the dimension D.

To this end, a determined number of hyperspheres is placed inside the first one, then, they are inflated. The inflation is stopped when the said hyperspheres are “kissing” each other. The search space E can be decomposed using a fractal dimension equal to twice its dimension D (2×D). Such decomposition allows covering most of the search space, and has a low complexity for computation. Thus, we can place a hypersphere on each side of each axis: two for each axis.

FIGS. 2a and 2b illustrate a hypersphere H approximating a search space with a dimension equal to 2. As represented, the hypersphere H^(k-1) is divided in 4 sub-hyperspheres sH^(k).

k is an integer corresponding to a deep (level) of the decomposition of the search space E.

Note that one refers either to a hypersphere H^(k) or a sub-hypersphere sH^(k), as it describes the same hypersphere at a given level k of fractal decomposition.

Thus, FIG. 2a illustrates a first level decomposition (k=1) of a first hypersphere H⁰ which is divided in 4 sub-hyperspheres sH¹.

FIG. 2b illustrates a second level of decomposition, where each hypersphere of the first level are divided in 4 sub-hyperspheres sH².

The choice of this number of sub-hyperspheres sH^(k) is guided by the following rule: the ratio between radii of the first hypersphere H^(k-1) and that of other 2×D sub-hyperspheres sH^(k) inside, is about to 1+√{square root over (2)}≈2.41.

In this case, the ratio does not depend on the dimension of the problem. Then, centers {right arrow over (C)}^(k), and the radius r_(k) of a hypersphere H^(k) at a given level k of fractal decomposition, are given by:

${\overset{\rightarrow}{C}}^{k} = \left\{ \begin{matrix} {{\overset{\rightarrow}{C}}^{k - 1} + {\left( {r_{k - 1} - \left( {r_{k + 1}\text{/}2.41} \right)} \right) \times {\overset{\rightarrow}{e}}_{j}}} & {,{{{{if}\mspace{14mu} i} = {2 \times j}};}} \\ {{\overset{\rightarrow}{C}}^{k - 1} - {\left( {r_{k - 1} + \left( {r_{k - 1}\text{/}2.41} \right)} \right) \times {\overset{\rightarrow}{e}}_{j}}} & {{,{{otherwise}.}}\mspace{11mu}} \end{matrix} \right.$

where {right arrow over (e)}_(j) is the unit vector at the dimension j is set to 1, and i is the number of hypersphere at a given level.

As shown illustrated in FIG. 3a , the decomposition of an hypersphere H using sub-hyperspheres sH for 2×D, does not entirely cover the search space E. Thus, the global optimum can be missed and the algorithm may not find the global optimum. To avoid this situation, in a step 130, an inflation factor of the obtained sub-hyperspheres is used. Radii of the sub-hyperspheres are increased by δ (its value was fixed to at least 1.75 empirically) to obtain inferior hyperspheres, represented with their centers and their radii. This enlargement allows overlaps between sub-hyperspheres sH to cover all the search space E as shown in FIG. 3 b.

Then, the inflated radii of hyperspheres are given by:

r _(k) =δ×r _(k-1)/2.41

where δ is the inflation coefficient, r_(k-1) and r_(k) are the radii of the hyperspheres at the level k−1 and r_(k), respectively.

Advantageously, δ is at least about 2.80, when the fractal dimension is equal to 2×D, as this coefficient is optimum.

A method of estimation of the lower bound value of the inflation coefficient is detailed further in the annex section.

Search of the Best Direction

Then, in a step 140 each sub-hypersphere sH at a k-level is evaluated to have its quality measure. Based on this fitness, these sub-hyperspheres are sorted, and the one with the best one is chosen to be the next hypersphere to be visited (decomposed). This procedure allows the method 100 to direct the search for the most promising region, and to lead the optimization within a reduced space at each level.

To this end, for each sub-hypersphere sH at a k-level, the initial solution {right arrow over (S)} is positioned at the center {right arrow over (C)}^(k) of the current hypersphere H (hypersphere to evaluate). Then, starting from the center of the current hypersphere H, two solutions {right arrow over (S)}₁ and {right arrow over (S)}₂ along each dimension of the search space are generated using the radius r_(k) as expressed previously.

Then, the best among these three solutions will represent the quality or fitness of the evaluated hypersphere.

${{\overset{\rightarrow}{s}}_{1} = {{\overset{\rightarrow}{C}}^{k} + {\frac{r_{k}}{\sqrt{D}} \times {\overset{\rightarrow}{e}}_{d}}}},{{{for}\mspace{14mu} d} = 1},2,\ldots \;,D$ ${{\overset{\rightarrow}{s}}_{2} = {{\overset{\rightarrow}{C}}^{k} - {\frac{r_{k}}{\sqrt{D}} \times {\overset{\rightarrow}{e}}_{d}}}},{{{for}\mspace{14mu} d} = 1},2,\ldots \;,D$

where {right arrow over (e)}_(d) is the unit vector which the d^(th) element is set to 1 and the other elements to 0, while, k is the current deep of decomposition. Then, for positions {right arrow over (S)}₁ and {right arrow over (S)}₂ and {right arrow over (S)} (positioned at the center of the current hypersphere {right arrow over (C)}^(k)), their fitnesses, f1, f2 and fc, respectively, are calculated as well as their, corresponding distances to the best position found so far (BSF) via the Euclidean distance. The last step consists of computing the slope at the three positions {right arrow over (S)}₁, {right arrow over (S)}₂ and {right arrow over (C)}^(k)), referred as g1, g2 and gc. This is performed by taking the ratio between the fitness (f1, f2 and fc) and their corresponding distances. Then, the quality for the current hypersphere will be represented by the highest ratio among g1, g2 and gc, denoted by q: q=max {g1; g2; gc} with:

${{g\; 1} = \frac{f\left( {\overset{\rightarrow}{S}}_{1} \right)}{{{\overset{\rightarrow}{S}}_{1} - {BSF}}}},{{g\; 2} = \frac{f\left( {\overset{\rightarrow}{S}}_{2} \right)}{{{\overset{\rightarrow}{S}}_{2} - {BSF}}}},{{{and}\mspace{14mu} {gc}} = {\frac{f\left( \overset{\rightarrow}{C} \right)}{{{\overset{\rightarrow}{C}}^{k} - {BSF}}}.}}$

Thus, advantageously, the following algorithm represents the search of the best solution for a given hypersphere:

  Step 1) Initialization:  Step 1.1) Initialize the solution {right arrow over (S)} at the center {right arrow over (C)}^(k) of the current  hypersphere H^(k).  Step 1.2) Evaluate the objective function of the solution {right arrow over (S)}. Step 2) Generation of solutions {right arrow over (S)}₁and {right arrow over (S)}₂:  For each dimension i do    ${\left. {{Step}\mspace{14mu} 2.1} \right)\mspace{14mu} s\; {1\lbrack i\rbrack}} = {{s\lbrack i\rbrack} - \frac{r_{k}}{\left. \sqrt{}D \right.}}$    ${\left. {{Step}\mspace{14mu} 2.2} \right)\mspace{14mu} s\; {2\lbrack i\rbrack}} = {{s\lbrack i\rbrack} + \frac{r_{k}}{\left. \sqrt{}D \right.}}$  End for Step 3) Evaluate the objective function of the solutions {right arrow over (S)}₁and {right arrow over (S)}₂. Step 4) Return the best solution in the set {{right arrow over (S)}, {right arrow over (S)}₁, {right arrow over (S)}₂}.

At a given level k, all hyperspheres can be stored in a sorted list Ik depending on their scores (fitnesses). Afterwards, the obtained list can be pushed into a data structure (for example a stack s_(memory) or a graph) to save regions that are not visited yet.

Alternatively, the search for a best solution for said given hypersphere can be performed through the implementation of an existing optimization software of the art. Such software can be selected from, for example, CPLEX (IBM), Gurobi, GLPK, MOSEK, CBC, CONOPT, MINOS, IPOPT, SNOPT, KNITRO, LocalSolver and CP Optimizer.

Once the search in the current hypersphere is terminated without success, in a step 150 the first hypersphere at the top of Ik (the hypersphere with the best quality) is selected to be decomposed and so on, by applying recursively the steps 120 to 150, until reaching the last level.

Once the last level of recursive decomposition, called the fractal depth k_(max), is reached, in a step 160, an intensive local search procedure (ILS) is applied to the sorted sub-hyperspheres to find the global optimum.

Advantageously, the fractal depth k_(max) is equal to 5. Such value allows a good compromise between the size of the space and a computation time.

Intensive Local Search (ILS)

At the last level k_(max), any stochastic or deterministic optimization method can be used. Herein, the goal of this work is to design a low complex and deterministic optimization method, consequently, a local search method based on variable neighbor search (VNS) strategy can be considered.

Indeed, the used local search is based on a line search strategy, and adapted to solve high-dimensional problems. The main idea is to consider a neighborhood search for each dimension of the space sequentially: at each iteration, the search is performed on only one dimension d. Thus, from the current solution {right arrow over (S)}, two solutions {right arrow over (S)}₁ and {right arrow over (S)}₂ are generated by moving {right arrow over (S)} along the dimension d using the equations given by:

{right arrow over (S)} ₁ ={right arrow over (S)}+γ×{right arrow over (e)} _(d)

{right arrow over (S)} ₂ ={right arrow over (S)}−γ×{right arrow over (e)} _(d)

where {right arrow over (e)}_(d) is the unit vector which the d^(th) element is set to 1 and the other elements are set to 0, and γ the step size is initialized to the value of the radius of the current hypersphere.

Then, the best among three solutions {right arrow over (S)}, {right arrow over (S)}₁, and {right arrow over (S)}₂ is selected to be the next current position {right arrow over (S)}.

The parameter γ is updated, only, if there is no improvement after checking in all dimensions.

Depending on the situation, the step size is adapted using the following rules:

-   -   if a better candidate solution was found in the neighborhood of         {right arrow over (S)}, then, the step size γ is for example         halved;     -   the step size γ is decreased until reaching a tolerance value         (γ_(min)), representing the tolerance or the precision searched.         Thus, the stopping criterion for the ILS is represented by the         tolerance value γ_(min). The value of γ_(min) is for example         fixed to 1×e⁻²⁰.

The ILS is executed without any restriction on the bounds. In other terms, the method 100 allows following a promising direction outside of the current sub-region bounds.

If the generated solution is outside the current inflated hypersphere, it is ignored and is not evaluated.

The following algorithm describes some steps of the ILS procedure:

Step 1) Initialization: Step 1.1) Initialize the current position {right arrow over (S)} at the center {right arrow over (C)}^(k) and define Υ_(min) of the current hypersphere H^(k). Step 1.2) Evaluate the objective function of the solution {right arrow over (S)}. Step 1.3) Initialize the step Υ with the radius r_(k) of the current hypersphere H^(k). Step 2) Neighbor search {right arrow over (S)}: For each dimension d do Step 2.1) s1[d] = s[d] − Υ Step 2.2) s2[d] = s[d] + Υ Step 2.3) Evaluate the objective functions of the solutions {right arrow over (S)}₁ and {right arrow over (S)}₂. Step 2.4) Update {right arrow over (S)} using the current position with the best one among {{right arrow over (S)}, {right arrow over (S)}₁, {right arrow over (S)}₂}. End for Step 3) If there is no improvement of {right arrow over (S)}, decrease the step Υ: Υ = Υ × 0.5. Step 4) Repeat Step 2-Step 3 until Υ < Υ_(min). Step 5) Return the best solution {right arrow over (S)}.

Alternatively, ILS can be performed using an existing optimization software as listed above.

When all of these sub-hyperspheres sH^(k) from the last level k_(max) are visited, in a step 170, the search is raised up to the previous level k−1, replacing the current hypersphere H with the following sub-hypersphere sH^(k) from the sorted sH list.

When all the plurality of hyperspheres issued from the k−1 level hypersphere are visited, it is removed from the data structure (for example a stack or a graph) where the hyperspheres not visited are stored.

Then, the steps 120 to 170 are repeated until the stopping criterion is reached or the sub-hyperspheres from all the levels are visited. In a step 180, the best solution among all the hyperspheres visited is returned by the method 100.

As we can notice in the proposed approach, all the mechanisms used are exact methods which denotes of its accuracy. Besides, sH ranking is important for the efficiency of the method 100, it favors the search in the most promising region to reach the global optimum faster.

Overview of the Method 100

The optimization process starts with hypersphere inside the problem search space, which is divided into a plurality of sub-regions (for example 2×D) delimited by the new smaller hyperspheres. This procedure is repeated until a given deep or level. At each level, the plurality of sub-regions are sorted regarding their fitness respectively, and only the best one is decomposed using the same procedure. However, the other sub-regions are not discarded, they would be visited later: if the last level search (ILS) is trapped in a local optimum.

As the different centers of the sub-regions were stored, when all sub-regions of the last level k were visited, and the global optimum was not found, the second sub-region at level k−1 is decomposed, and the ILS procedure is started again from the best sub-region. In some cases (i.e. optimal solution at the limit of the search space), the ILS algorithm can generate positions out of the search space, then, they are simply ignored. The diversification (going from a hypersphere to another at different levels) and intensification procedures (last level search) pointed out may lead to a deeper search within a subspace which is geometrically remote from the subspace selected for re-dividing at the current level.

The following algorithm illustrates the succession of the different steps of decomposition, evaluation of quality and intensive local search of the method 100.

Step 1) Initialization of the hypersphere H: Step 1.1) Initialize the center C at the center of the search space. Step 1.2) Initialize the radius R with the distance between the center C and the lower bound l or the upper bound U of the search space. Step 2) Divide the hypersphere H using the Search Space Fractal Decomposition method. Step 3) Use of the local search: For each sub-hypersphere sH do Step 3.1) Evaluate the quality of the hypersphere sH. End for Step 4) Sort the sub-hyperspheres sH using the local optima found for each one. Step 5) Update the current hypersphere H: If the last level is reached then For each sub-hypersphere sH do Apply the ILS. End for Step 5.1) Replace the current hypersphere H with the next sub-hypersphere sH from the previous level. Else Step 5.2) Replace the current hypersphere H with  the best sub-hypersphere sH. End if Step 6) Repeat Step 2-Step 5 until a stopping criterion is met.

One can notice that the proposed algorithm is deterministic: the last level heuristic is exact and the decomposition also does not contain any stochastic process.

Advantageously, the method 100 can be implemented in parallel computation architectures like cloud computing, clusters, HPC etc., as the method is particularly adapted to such architecture. Indeed, the processing of steps of determination of quality and/or last level search can be implemented in separated threads for each sub-hypersphere being processed.

Complexity Analysis

The proposed method 100 includes three distinct parts. The first one is the decomposition of hyperspheres process, the second corresponds to the quality evaluation of the hypersphere and the third is the last level local search.

Then the complexity of the method 100 is given by the following table:

Asymptotic complexity Decomposition of hyperspheres log_(k)(D) Quality evaluation of an hypersphere 1 Local search log₂(r/Υ_(min))) where D represents the problem dimension D, r the radius of the current hypersphere and γ_(min) the local search tolerance threshold. Thus, the method 100 has a complexity given by: O(log_(k)(D)+1+log 2(r/γ_(min)))=O(log_(k)(D)).

Hence, the asymptotic complexity of the proposed approach is O(log_(k)(D)). Thus, the method 100 has a logarithmic complexity depending on deep of the search.

On the other hand, the proposed algorithm uses a stack to store the hyperspheres that are not visited yet. Indeed, after each decomposition, the list of the obtained hyperspheres is pushed, for example, into a stack implemented for this purpose, except for the last level. Once a hypersphere is entirely visited, it is removed from the list of the current level, and selects the next one to be processed. Besides, in the case of 2×D decomposition, for the current list of hyperspheres, 2×D solutions representing their qualities are also stored in a list in order to sort the hyperspheres.

In our case, in the worst case, the lists concerning the k−1 levels contains 2×D hyperspheres which makes the algorithm use (k−1)×(2×D)+2×D of work memory. Thus, the memory complexity of the method 100 is 0(D).

Moreover, to store the list of hyperspheres, we can only store the coordinates of the center of the hyperspheres, thus allowing a low consumption of memory.

Thus, there is no need to save all information about visited hyperspheres by the proposed method: all decomposition can be reconstructed analytically. Then, only the best positions met are saved. We can compute the center positions C of the hyperspheres H without any past position.

Hence, alternatively, the center positions C of the hyperspheres H of a given level k can be deduced from the center position C of a given hypersphere H of the said level. In that case, only one hypersphere H is stored and coordinates of the center of the other hyperspheres H are not stored, thus allowing less memory usage. This is particularly advantageous when the method 100 is implemented on small computers as for example, microcontrollers, embedded systems or by using a Field-Programmable Gate Array (FPGA).

Performance Comparison

FIG. 4 illustrates the ranking of 16 algorithms taken from the literature ([1]-[9]) of the proposed approach (FDA) for dimensions D=50, D=100, D=200, D=500 and D=1000 using a radar plot. The used rank method is based on the Friedman and Quade tests [10]. In fact, the lower the rank is, the better the algorithm is. Hence, the illustration shows that the proposed approach is competitive with the mDE-bES and SaDE and far better than the other compared algorithms.

FIGS. 7a to 7e show boxplots illustrating a distribution comparison of average ranks between the method according to the invention and known algorithms for set of dimensions respectively equal to 50, 100, 200, 500 and 1000. In those plots, the circle highlights the outliers, meaning the functions where the algorithm performs surprisingly good or bad. In our case, it is clear that the proposed approach (FDA) shows better and stable performance among all dimensions on all functions.

Application of the Method 100 to Hydroelectric Benchmark

FIG. 5 illustrates an optimization problem of a hydroelectric power station 500.

The problem is to optimize the production of electricity given multiple constraints.

The constraints on a pipe P connecting a reservoir to an End are as follows:

-   -   a reserved flow of 2 m³/s is imposed throughout the year         (q^(LO)=2 m³/s);     -   there is no maximum limitation of flood flow;     -   no flow gradient is imposed;     -   transfer time: δ_(tn), =1 hour.

The constraints on the reservoir tank are as follows:

-   -   the minimum V^(LO) volume is m³;     -   the maximum V^(UP) volume is 25,000,000 m³;     -   no pipe reaches this tank;     -   input water is required;     -   3 pipes exit: one towards the turbine T1, one towards the         turbine T2 and one towards the End;     -   the tank will not overflow unless it is possible to do         otherwise;     -   the initial volume is equal to the maximum volume of the tank;     -   the final volume is equal to the initial volume.

The constraints on the turbine T1 in a linear case are as follows:

-   -   the maximum turbulent flow rate is: q^(UP)=230 m³/s;     -   production sold at spot prices;     -   production curve=1.07*Flow rate.

The constraints on the turbine T2 in a linear case are as follows:

-   -   the maximum turbulent flow rate is: q^(UP) 180 m³/S;     -   production sold at spot prices;     -   production curve=1.1*debit.

The constraints on the turbine T1 in a non-linear case are as follows:

-   -   the maximum turbine flow rate is: q^(UP)=230 m³/s     -   minimum technical operation: fLO=27 MW;     -   production sold at spot prices;     -   production curve=f (Flow, Volume);     -   optional: constraint that the plant runs at least 2 consecutive;     -   optional: the plant can only start up to 3 times on the same         day.

The constraints on the turbine T2 in a non-linear case are as follows:

-   -   the maximum turbinable flow rate is: q^(UP)=180 m³/s;     -   minimum technical operation: fLO=135 MW;     -   production sold at spot prices;     -   production curve=f (Flow, Volume);     -   optional: the plant must stop at least 3 hours before it can be         restarted;     -   optional: the T2 plant must not operate if the tank of the         reservoir is below 3,350,000 m³.

The following table shows the comparison between the optimization by the method 100 and by the software Cplex (Ibm) given different scenarios and in the case of a linear problem.

As we can observe, the method 100 allows finding the same global optimum as other known methods.

Scenario HydroKISS Cplex sc1 3.37E+07 3.35E+07 sc2 4.67E+07 4.67E+07 sc3 4.79E+07 4.80E+07 sc4 4.45E+07 4.46E+07 sc5 3.09E+07 3.09E+07 sc6 3.39E+07 3.39E+07 sc7 4.04E+07 4.05E+07 sc8 4.22E+07 4.23E+07 sc9 4.22E+07 4.23E+07 sc10  3.38E+07 3.38E+07

The following table shows the comparison between the optimization by the method 100 and by the software Cplex (IBM) given different scenarios (each scenario corresponding to specific electricity prices and water income constraints) and in the case of a non-linear problem.

As we can observe, contrary to the Cplex method which cannot converge in all the scenarios (except sc1), the method 100 allows determining a global optimum.

Scenario HydroKISS Cplex-Repaired sc1 3.50E+07 2.83E+07 sc2 4.01E+07 — sc3 4.25E+07 — sc4 3.93E+07 — sc5 2.66E+07 — sc6 3.04E+07 — sc7 3.49E+07 — sc8 3.49E+07 — sc9 3.70E+07 — sc10  2.94E+07 —

Application of the Method 100 to Image Registration

The performance of the optimization method 100 can be illustrated on a 2D image registration.

The image registration is a technic widely spread in the medical imaging analysis, its goal is to determine the exact transformation which occurred between two images of the same view.

In first, a brief description of the image registration process is presented, and formulated as an optimization problem.

An example of a 2D image of coronary arteries obtained with an X-ray angiography can be considered.

The idea is to perform transformation on one of the two images until a defined similarity measure is optimized. Registration can be used, for instance, to detect the speed growth of a tumor, or study the evolution of the size and shape of a diseased organ.

The registration method follows the pattern herein after:

-   -   First, determination of the type of the transformation which         occurred between the two images. There are a large number of         possible transformations described in the literature and the         model choice is an input to the registration system (procedure).         Herein, a rigid transformation was used.     -   Then, the similarity measure to use: we use the quadratic error.         This criterion must be minimized to find the optimal parameters         of the transformation between the two images. Let I1 and I2 be         the two images to be compared composed of N pixels. I1 (X) is         the intensity of the pixel X in the image 1. The quadratic error         E (ϕ) becomes: E(Φ)=Σ_(X=1) ^(N)[I1(X)−I2(T(X))]² where T is the         transformation for a given ϕ.     -   −To minimize the previous criterion, an optimization method has         to be used. Fractal approach will be used to solve this problem.

It is very practical to use this transformation for illustration. A rigid transformation is defined by two things: a rotation of an angle) around a point and a translation T (a, b).

FIG. 6a illustrates such transformation and the use of the method 100 to determine its parameters.

An image 13 is the result of the image I2 with the transformation T. FIG. 6b illustrates a convergence curve of the errors, between the two images I1 and 13, when applying the method 100 to optimize the parameters of the transformation T.

At final, the two images I1 and 13 are the same with an error of 10⁻⁷.

Thus, the method 100 allows finding the best parameters to register the images.

Application of the Method 100 to the Problem of Molecular Aggregation

The Lennard-Jones (U) problem consists, given an aggregate of K atoms, representing a molecule, in finding the relative positions of these atoms whose overall potential energy is minimal in a three-dimensional Euclidean space.

The problem LJ is as follows: the molecule consists of K atoms, x^(i)={x^(i) ₁, x^(i) ₂, x^(i) ₃) represents the coordinates of the atom i in three-dimensional space, knowing that the K atoms are located at positions ((x¹), . . . , (x^(k))) where x^(i)∈R³ and i=1, 2, . . . , K. The potential energy formula of U for a pair of atoms is given by:

${v\left( r_{ij} \right)} = {\frac{1}{r_{ij}^{12}} - \frac{1}{r_{ij}^{6}}}$

such that 1≤i, j≤K where Vij=∥x^(i)−x^(j)∥.

The total potential energy is the sum of the energies resulting from the interactions between each couple of atoms of the molecule. The most stable configuration of the molecule corresponds to an overall minimum potential energy that can be obtained by minimizing the function V_(k)(x) in the following equation:

${{Min}\mspace{14mu} {V_{k}(x)}} = {{\Sigma_{i < j}{v\left( {{x_{i} - x_{j}}} \right)}} = {\sum\limits_{i = 1}^{K - 1}\; {\sum\limits_{j = {i + 1}}^{K - 1}\; \left( {\frac{1}{{{x_{i} - x_{j}}}^{12}} - \frac{1}{{{x_{i} - x_{j}}}^{6}}} \right)}}}$

For example, we use 9 atoms in a research space [−2, 2]. The value of the overall optimum in this case is f*=−24,113360.

The other optimization algorithm that was used to solve this problem is the SPSO 2007 (Standard Particle Swarm Optimization). The dimension of the problem is 27, this problem is not large scale but this comparison shows that even on small dimensions the method 100 is adapted and competitive.

Number SPSO Method Global of atoms 2007 100 optimum 9 +2.31 −10.6855 −24.113360

REFERENCES

[1] R. Storn and K. Price, Differential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces. ICSI Berkeley, 1995, vol. 3.

-   [2] L. J. Eshelman and J. D. Schaffer, “Real-coded genetic     algorithms and interval-schemata.” in FOGA, L. D. Whitley, Ed.     Morgan Kaufmann, 1992, pp. 187-202. -   [3] A. Auger and N. Hansen, “A restart cma evolution strategy with     increasing population size”, in 2005 IEEE Congress on Evolutionary     Computation, vol. 2, September 2005, pp. 1769-1776 Vol. 2. -   [4] D. Molina, M. Lozano, A. M. Sanchez, and F. Herrera, “Memetic     algorithms based on local search chains for large scale continuous     optimization problems: Ma-ssw-chains”, Soft Computing, vol. 15, no.     11, pp. 2201-2220, 2011. -   [5] L.-Y. Tseng and C. Chen, “Multiple trajectory search for large     scale global optimization”, in 2008 IEEE Congress on Evolutionary     Computation (IEEE World Congress on Computational Intelligence),     June 2008, pp. 3052-3059. -   [6] A. LaTorre, S. Muelas, and J.-M. Pena, “A mos-based dynamic     memetic differential evolution algorithm for continuous     optimization: a scalability test”, Soft Computing, vol. 15, no. 11,     pp. 2187-2199, 2011. -   [7] A. K. Qin and P. N. Suganthan, “Self-adaptive differential     evolution algorithm for numerical optimization”, in 2005 IEEE     Congress on Evolutionary Computation, vol. 2, September 2005, pp.     1785-1791 Vol. 2. -   [8] A. Duarte, R. Marti, and F. Gortazar, “Path relinking for     large-scale global optimization”, Soft Computing, vol. 15, no. 11,     pp. 2257-2273, 2011. -   [9] M. Z. Ali, N. H. Awad, and P. N. Suganthan, “Multi-population     differential evolution with balanced ensemble of mutation strategies     for large-scale global optimization”, Applied Soft Computing, vol.     33, pp. 304-327, 2015. [Online]. Available:     http://www.sciencedirect.com/science/article/pii/S1568494615002 458 -   [10] E. Theodorsson-Norheim, “Friedman and Quade tests: Basic     computer program to perform nonparametric two-way analysis of     variance and multiple comparisons on ranks of several related     samples”, Computers in biology and medicine, vol. 17, no. 2, pp.     85-99, 1987.

Annex: Inflation Coefficient Estimation

Considering the radius r of outer hypersphere H and r′ the radius of sub-hyperspheres sH: let the centers of sub-hyperspheres sH be located in points 0_(i) ^(±)=(0, . . . , 0, ^(±)d, 0, . . . , 0); where the only non-zero entry ^(±)d is in i-th position, and

$d = {{r - r^{\prime}} = {{r\left( {1 - \frac{1}{1 + \left. \sqrt{}2 \right.}} \right)}.}}$

The minimal required inflation rate δ_(n) is estimated such as any arbitrary point from inside of a hypersphere H is covered by one of inflated sub-hyperspheres sH with radius r″=δ_(n) r′. To this end, first we estimate the inflated radius r_(A) for an arbitrary fixed point A; and then r″ is defined as r″=max r_(A).

Let point A(x₁, . . . , x_(n)) lie inside the hypersphere H, i.e. x₁ ²+x₂ ²+ . . . +x_(n) ²≤r². Radius r_(A) is minimal required to cover A, thus

$\begin{matrix} {r_{A}^{2} = {{\min\limits_{i, \pm}\mspace{14mu} {{O_{i}^{\pm}A}}^{2}} = {\min\limits_{i, \pm}\mspace{14mu} \left( {{\sum\limits_{j \neq i}x_{j}^{2}} + \left( {x_{i} \pm d} \right)^{2}} \right)}}} \\ {= {\min\limits_{i, \pm}\mspace{14mu} {\left( {{\sum\limits_{j \neq i}x_{j}^{2}} + \left( {{x_{i}} \pm d} \right)^{2}} \right).}}} \end{matrix}$ As  x_(i) ≥ 0, d > 0, then  (x_(i) − d)² < (x_(i) ± d)².  Hence $r_{A}^{2} = {\min\limits_{i}\mspace{14mu} {\left( {{\sum\limits_{i \neq j}x_{j}^{2}} + \left( {{x_{i}} - d} \right)^{2}} \right).}}$

Denote

$a^{2} = {{{OA}}^{2} = {\sum\limits_{i = 1}^{n}{x_{i}^{2}.}}}$

Then

$\begin{matrix} {r_{A}^{2} = {\min\limits_{i}\left( {{\sum\limits_{i \neq j}x_{j}^{2}} + \left( {{x_{i}} - d} \right)^{2}} \right)}} \\ {= {\min\limits_{i}\left( {a^{2} - x_{i}^{2} + \left( {{x_{i}} - d} \right)^{2}} \right)}} \\ {= {a^{2} + {\min\limits_{i}\left( {{- x_{i}^{2}} + \left( {{x_{i}} - d} \right)^{2}} \right)}}} \\ {= {a^{2} - {\max\limits_{i}\left( {x_{i}^{2} - \left( {{x_{i}} - d} \right)^{2}} \right)}}} \\ {= {a^{2} - {\max\limits_{i}{d\left( {{2{x_{i}}} - d} \right)}}}} \\ {= {a^{2} - {d\; {\max\limits_{i}\left( {{2{x_{i}}} - d} \right)}}}} \\ {= {a^{2} - {{d\left( {{2{\max\limits_{i}{x_{i}}}} - d} \right)}.}}} \end{matrix}$

Then, in order to cover all points inside hypersphere we take the maximum of inflation radius r_(A) over A. Thus

$\begin{matrix} {r^{''2} = {\max\limits_{{{OA}}^{2} \leq r^{2}}\mspace{14mu} r_{A}^{2}}} \\ {= {\max\limits_{0 \leq a \leq r}\mspace{14mu} {\max\limits_{{{OA}} = a}\mspace{14mu} \left( {a^{2} - {d\left( {{2\mspace{14mu} {\max\limits_{i}\mspace{14mu} {x_{i}}}} - d} \right)}} \right)}}} \end{matrix}$

We show that point A cannot be a maximum point unless all of its coordinates are equal up to the sign. Without loss of generality assume that x_(i)>0 for all 1≤i≤n (indeed, the sign of x_(i) does not influence the value of considered function under max operator). Assume that there exists an index s such that x_(s)≠max x_(k). Let i₁, i₂, . . . , i_(n) be the indices of the biggest coordinates of A, and j be the entry of the second biggest coordinate:

$x_{i_{1}} = {x_{i_{2}} = {\ldots = {x_{i_{m}} = {\max\limits_{k}\mspace{14mu} x_{k}}}}}$ $x_{j} = {\max\limits_{{k \neq i_{1}},i_{2},\ldots,i_{m}}\mspace{14mu} x_{k}}$

Under this assumption it is possible to “shift” point A along the sphere ∥OA∥=a in such a way that the maximized function is increasing, and thus point A cannot be a point of maximum. Consider another point, A′, with coordinates (x′1, . . . , x′n) where

x _(k) ′=x _(k) , k≠i ₁ , i ₂ , . . . , i _(m) ,j,

x _(i) ′=x _(i)−δ⁻ ,|i=i ₁ ,i ₂ , . . . ,i _(m),

x _(j) ′=x _(j)+δ⁺,

and 0<δ⁻,δ⁺<½(x _(i) ₁ −x _(j)) are such that ∥OA′∥ ² =∥OA∥ ²=α². Then

x _(i) ′=x _(i)−δ⁻ >x _(j)+δ⁺ =x _(j) ′,i=i ₁ ,i ₂, . . . ,

x _(j) ′=x _(j)+δ⁺ >x _(j) ≥x _(k) =x _(k) ′,k≠i ₁ ,i ₂ , . . . ,i _(m) ,j.

Hence

$x_{i_{1}}^{\prime} = {x_{i_{2}}^{\prime} = {{\ldots \; x_{i_{m}}^{\prime}} = {\max\limits_{k}\mspace{14mu} {x_{k}^{\prime}.}}}}$

Denote

${{f(A)} = \left( {a^{2} - {d\left( {{2\; {\max\limits_{k}{x_{k}}}} - d} \right)}} \right)},$

where as before α²=∥OA∥². Then

$\begin{matrix} {{f\left( A^{\prime} \right)} = \left( {a^{2} - {d\left( {{2x_{i_{1}}^{\prime}} - d} \right)}} \right)} \\ {= \left( {a^{2} - {d\left( {{2x_{i_{1}}} - {2\delta^{-}} - d} \right)}} \right)} \\ {= {{\left( {a^{2} - {d\left( {{2x_{i_{1}}} - d} \right)}} \right) + {2\delta^{-}d}} > {{f(A)}.}}} \end{matrix}$

In other words,

${\underset{{{OA}} = a}{argmax}\left( {a^{2} - {d\left( {{2{\max\limits_{i}{x_{i}}}} - d} \right)}} \right)} \neq {A.}$

We can also remark here that the maximum exists. Indeed, function f(A) is continuous and can be considered as function on a compact of lower dimensions, defined by equality ∥OA∥=a. Hence by extreme value theorem it reaches its maximum.

Basing on the inequality above, we infer that the point of maximum has to have equal coordinates (up to the sign). In particular, the maximum is reached in point A* with coordinates

$\left( {\frac{a}{\sqrt{n}},\ldots \mspace{20mu},\frac{a}{\sqrt{n}}} \right).$

Returning to the expression for inflation radius r″; we get

$\begin{matrix} {r^{''2} = {\max\limits_{0 \leq a \leq r}{\max\limits_{{{OA}} = a}\left( {a^{2} - {d\left( {{2{\max\limits_{i}{x_{i}}}} - d} \right)}} \right)}}} \\ {= {\max\limits_{0 \leq a \leq r}\left( {a^{2} - {d\left( {\frac{2a}{\sqrt{n}} - d} \right)}} \right)}} \\ {= {\max\limits_{0 \leq a \leq r}{\left( {a^{2} - \frac{2{ad}}{\sqrt{n}} + d^{2}} \right).}}} \end{matrix}$

The quadratic function of a, under the max operator, reaches its maximum on one of interval ends. Denote

${g(a)} = {a^{2} - \frac{2{ad}}{\sqrt{n}} + {d^{2}.}}$

Then

${{g(0)} = {d^{2} = {\lambda^{2}r^{2}}}},{{g(r)} = {{r^{2} - \frac{2r\; d}{\sqrt{n}} + d^{2}} = {r^{2} - \frac{2\lambda r^{2}}{\sqrt{n}} + {\lambda^{2}r^{2}}}}},$

where

$\lambda = {{1 - \frac{1}{1 + \sqrt{2}}} = {\frac{\sqrt{2}}{1 + \sqrt{2}} \approx {0.59.}}}$

From this, it shows that for n≥2

$\begin{matrix} {{\frac{2\lambda}{\sqrt{n}} < 1},{{{and}\mspace{14mu} {so}\mspace{14mu} {g(r)}} > {{g(0)}.}}} & \; \\ {{Finally},\begin{matrix} {r^{''\; 2} = {\max\limits_{0 \leq a \leq r}\left( {a^{2} - \frac{2{ad}}{\sqrt{n}} + d^{2}} \right)}} \\ {= {g(r)}} \\ {= {r^{2} - \frac{2\lambda \; r^{2}}{\sqrt{n}} + {\lambda^{2}r^{2}}}} \\ {{= {r^{2}\left( {1 - \frac{2\lambda}{\sqrt{n}} + \lambda^{2}} \right)}},} \end{matrix}} & \; \end{matrix}$

And thus

$\begin{matrix} {\alpha_{n} = {\frac{r^{''}}{r^{\prime}} = {\frac{r\sqrt{1 - \frac{2\lambda}{\sqrt{n}} + \lambda^{2}}}{{r\left( {1 + \sqrt{2}} \right)}^{- 1}} = {\left( {1 + \sqrt{2}} \right)\sqrt{1 - \frac{2\lambda}{\sqrt{n}} + \lambda^{2}}}}}} \\ {= {\left( {1 + \sqrt{2}} \right)\sqrt{1 - \frac{2\sqrt{2}}{\sqrt{n}\left( {1 + \sqrt{2}} \right)} + \frac{2}{\left( {1 + \sqrt{2}} \right)^{2}}}}} \\ {= \sqrt{\left( {1 + \sqrt{2}} \right)^{2} - \frac{2\sqrt{2}\left( {1 + \sqrt{2}} \right)}{\sqrt{n}} + 2}} \\ {= {\sqrt{5 + {2\sqrt{2}} - \frac{2\sqrt{2}\left( {1 + \sqrt{2}} \right)}{\sqrt{n}}}.}} \end{matrix}$

In conclusion, in case we set the unified inflation coefficient, it would have to be equal to:

$\alpha = {\sup\limits_{n \geq 1}\alpha {_{n}{= {{\sup\limits_{n \geq 1}\sqrt{5 + {2\sqrt{2}} - \frac{2\sqrt{2}\left( {1 + \sqrt{2}} \right)}{\sqrt{n}}}} = {\sqrt{5 + {2\sqrt{2\;}}} \approx 2.80}}}}}$ 

1. A computer implemented method to optimize operation of a technical system by solving deterministically a nonlinear optimization problem implying technical constraints relating to technical parameters under which the said technical system operates, the technical parameters being of a number greater than 50, wherein the method comprises fractal geometric partitioning of a search space into a plurality of hyperspheres as a geometrical unitary pattern and wherein the hyperspheres partitioning said search space are overlapping; calculating a quality for each hypersphere; selecting from the plurality of hyperspheres, the hypersphere with the best quality; and further comprises determining an optimum solution of the said selected hypersphere, the optimum solution comprising values of the technical parameters to be implemented in the said technical system.
 2. The computer implemented method to optimize operation of a technical system according to claim 1, said solving method comprising the following steps: a) Initializating a search hypersphere with a dimension D equal to the number of the technical parameters; b) decomposing said search hypersphere into a plurality of sub-hyperspheres, the said sub-hyperspheres overlapping each other; c) for each sub-hypersphere, calculating a sub-hypersphere quality and ranking of the sub-hyperspheres as a function of said calculated quality; d) determining among the sub-hyperspheres, the sub-hypersphere having the best quality; e) until a first stopping criterion is reached, repeating steps b) to d), implemented on the basis of a search hypersphere (H) corresponding to the sub-hypersphere having the best quality as determined in the previous step; f) when the first stopping criterion is reached, for each sub-hypersphere resulting from the last implementation of step b), determining and storing a solution of each sub-hypersphere into a memory area of the computing unit; g) until a second stopping criterion is reached, implementing steps b) to e) on the basis of a search hypersphere corresponding to the following sub-hypersphere of the classification determined in step c) following the penultimate implementation of b); and h) determining an optimum solution among the solutions stored in the memory area and storing the values of the technical parameters corresponding to this optimum solution.
 3. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the first stopping criterion is a maximum level of recursive decomposition of the initial search hypersphere.
 4. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the second stopping criterion is a tolerance threshold.
 5. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the second stopping criterion is reached when a solution is stored for all the sub-hyperspheres from all the decomposition levels of the fractal partitioning.
 6. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step b) comprises the decomposition of the search hypersphere in 2×D sub-hyperspheres.
 7. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step a) of initialization comprises the determination of a maximum level of recursive decomposition of the initial search hypersphere.
 8. The computer implemented method to optimize operation of a technical system according to claim 7, wherein the maximum level of recursive decomposition of the search hypersphere is equal to
 5. 9. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step c) further comprises storing sub-hypersphere classification into a memory area.
 10. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step a) of initialization of the search hypersphere comprises: determining the position of the center C and the initial radius R of the search hypersphere, such as C=L+(U−L)/2, R=(U−L)/2, wherein U is the upper bound and L is the lower bound of a search space.
 11. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step d) of determination among the sub-hyperspheres of the sub-hypersphere having the best quality comprises: for each sub-hypersphere (i) starting from the center {right arrow over (C)} of a sub-hypersphere, generating two solutions {right arrow over (S)}₁ and {right arrow over (S)}₂ along each dimension d of the search space (E) with ${\overset{\rightarrow}{S\; 1} = {\overset{\rightarrow}{C} + {\frac{r}{\sqrt{D}} \times \overset{\rightarrow}{e_{d}}}}},{{\overset{\rightarrow}{S\; 2} = {\overset{\rightarrow}{C} - {\frac{r}{\sqrt{D}} \times \overset{\rightarrow}{e_{d}}}}};}$ (ii) determining the quality q of the sub-hypersphere with q=max {g1; g2; gc}, wherein: ${{g1} = \frac{f\left( {\overset{\rightarrow}{S}}_{1} \right)}{{{\overset{\rightarrow}{S}}_{1} - {BSF}}}},{{g2} = \frac{f\left( {\overset{\rightarrow}{S}}_{2} \right)}{{{\overset{\rightarrow}{S}}_{2} - {BSF}}}},{{{and}\mspace{14mu} {gc}} = \frac{f\left( \overset{\rightarrow}{C} \right)}{{{\overset{\rightarrow}{C}}^{k} - {BSF}}}},$ BSF corresponds to the position of the sub-hypersphere with the best quality determined so far, and ƒ({right arrow over (S)}₁), ƒ({right arrow over (S)}₂), ƒ({right arrow over (C)}) correspond to the fitness of respectively {right arrow over (S)}₁, {right arrow over (S)}₂ and {right arrow over (C)}.
 12. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the sub-hyperspheres overlapping each other of step b) are obtained by decomposing the search hypersphere into a plurality of sub-hyperspheres, then applying an inflation factor of the said sub-hyperspheres.
 13. The computer implemented method to optimize operation of a technical system according to claim 12, wherein the inflation factor corresponds to an increase of the radius of sub-hyperspheres by a factor of at least 1.75.
 14. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the inflation factor corresponds to an increase of the radius of sub-hyperspheres by a factor of at least 2.80.
 15. The computer implemented method to optimize operation of a technical system according to claim 1, wherein a plurality of sub-hyperspheres is stored into a memory area.
 16. The computer implemented method to optimize operation of a technical system according to claim 15, wherein the plurality of sub-hyperspheres are stored into a memory area by the storing the position of the center C of the said sub-hyperspheres.
 17. The computer implemented method to optimize operation of a technical system according to claim 15, wherein the position of a plurality of sub-hyperspheres is computed from a position of a sub-hypersphere stored into a memory area.
 18. (canceled)
 19. The computer implemented method to optimize operation of a technical system according to claim 2, wherein the step d) determining the sub-hypersphere having the best quality and/or the step f) of determining and storing a solution of each sub-hypersphere into a memory area is performed using existing optimization software.
 20. A computer program product having computer-executable instructions to enable a computer system to perform the method of claim
 1. 